In complex dimension 1, (some) Riemann surfaces of odd genus admit a holomorphic involution without fixed-point. In complex dimension 2, abelian surfaces and K3 surfaces are the first examples that come to mind (for admitting a holomorphic involution without fixed-point). There are possibly other example that are elliptic fibrations over a curve.

I am looking for a larger pool of such varieties in complex dimensions 2 and 3.

• What are the restrictions imposed by the existence of a holomorphic involution without fixed-point? (For instance, rationally connected varieties do not admit such an involution)

• Are all examples in dim 2 and 3 abelian or K3 fibered varieties?